[[Category ring]]
# Modules over a category ring

Let $\cat C$ be a category with finite $\Ob(\cat C)$.
Then a [[Module over a unital associative algebra|module]] over the the [[Category ring|category ring]] $\mathbb{K} [\cat C]$ is equivalent to a [[functor]] $\cat C \to \Vect_{\mathbb{K}}$, #m/thm/rep
and we have an [[equivalence of categories]][^1] 
$$
\begin{align*}
\lMod{\mathbb{K}[\cat C]} \simeq {\Vect_{\mathbb{K}}}^{\cat C}.
\end{align*}
$$


> [!check]- Proof
> Let $F : \cat C \to \Vect_{\mathbb{K}}$ be a functor,
> Let $(VF)_{X} = FX$ and thus construct the $\Ob(\cat C)$-[[graded vector space]]
> $$
> \begin{align*}
> VF = \bigoplus _{X \in \Ob(\cat C)} (VF)_{X}
> \end{align*}
> $$
> and for a morphism $f \in \cat C$ and homogenous vector $v \in (VF)_{X}$ define
> $$
> \begin{align*}
> f \odot v = \begin{cases}
> (Ff)v & X = \opn{dom} f \\
> 0 & X \neq \opn{dom}f
> \end{cases}
> \end{align*}
> $$
> and extend this definition linearly first for nonhomogenous vectors and then general $f \in \mathbb{K}[\cat C]$.
> Clearly this defines a $\mathbb{K}[\cat C]$-module.
> 
> Now suppose $\varphi \in {\Vect_{\mathbb{K}}}^\cat{C}(F,G)$ is a natural transformation with components $\varphi_{X} : (VF)_{X} \to (VG)_{X}$.
> Then
> $$
> \begin{align*}
> (V\varphi) = \bigoplus _{X \in \opn{Ob}(\cat C)} \varphi_{X} : VF \to VG
> \end{align*}
> $$
> defines an $\Ob(\cat C)$-[[Homomorphism of graded vector spaces|graded linear map]].
> Moreover, by naturality of $\varphi$, for a morphism $f \in \cat C(X,Y)$ and homogenous vector $v \in (VF)_{X}$
> $$
> \begin{align*}
> (V\varphi)\,(f \odot v) = \varphi_{Y}(Ff) v = (Gf) \varphi_{X} v = f \odot  (V\varphi) v
> \end{align*}
> $$
> so by linearity $V\varphi$ is a $\mathbb{K}[\cat C]$-module isomorphism.
> Therefore $V : {\Vect_{\mathbb{K}}}^{\cat C} \to \lMod{\mathbb{K}[\cat C]}$ is a functor.
> 
> Conversely, suppose $V$ is a $\mathbb{K}[\cat C]$-module.
> We define a functor $MV : \cat C \to \Vect_{\mathbb{K}}$ as follows:
> 
> - $(MV)X = 1_{X} \odot V$ for $X \in \Ob(\cat C)$;
> - $((MV)f) v = f \odot v$ for $f \in \cat C(X,Y)$ and $v \in (MV)X$.
> 
> Now suppose $\varphi : V \to W$ is a $\mathbb{K}[\cat C]$-[[module homomorphism]].
> We define a transformation with components
> $$
> \begin{align*}
> (M\varphi)_{X} : (MV)X &\to (MW) X\\
> v &\mapsto \varphi v
> \end{align*}
> $$
> which is well-defined since $\varphi$ is $\Ob(\cat C)$-graded.
> Moreover, for any $f \in \cat C(X,Y)$ and $v \in M(V)X$
> $$
> \begin{align*}
> ((MW) f) \,(M\varphi)_{X} \,v = f \odot  \varphi v = \varphi(f \odot  v) = (M\varphi)_{Y}\,((MV)f)
> \end{align*}
> $$
> so the following diagram commutes
> 
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> 
> whence $M\varphi$ is natural and $M : \lMod{\mathbb{K}[\cat C]} \to {\Vect_{\mathbb{K}}}^{\cat C}$ is a functor.
> 
> It is not difficult to see the natural equivalences required to make this an equivalence. <span class="QED"/>

#
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#state/tidy | #lang/en | #SemBr

[^1]: assuming the [[Axiom of Choice]].